Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,12)(11,15,16,14)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32);(2,5)(3,7)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30);(1,2,4,5)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,10,13,12)(11,14,16,15)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=fgh,d^2=fg,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#50113, #7705278, #7705685, … |
| maximal normal subgroup of | C2×#50113, #7705278, #7705685, … |
| maximal quotient of | #560799, #560971, #561908, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43101);